Mathematical Analysis of the Complete Iterative Inversion Method II

Abstract

A gas composed of identical isotropic molecules has a potential energy of interaction between pairs of particles that depends only on their separation distance. The pair potential is encoded in the virial coefficients of the virial equation of state for a gas. The complete iterative inversion method (CIIM) is an algorithm employed in an attempt to recover the pair potential from the second virial coefficient through successive approximations. In an earlier investigation we identified a very general class of “admissible” pair potentials for which the implicit assumptions of the CIIM are valid: improper integrals converge, derivatives exist, etc. Furthermore, we showed that the CIIM cannot recover the pair potential even if the target potential and the initial estimate are infinitely differentiable. For analytic pair potentials, it is known that the second virial coefficient uniquely determines the potential. The present work represents significant progress in the development of the mathematical framework suitable for confirming this uniqueness result for admissible analytic potentials within the universe of discourse of the CIIM. In particular, we formulate the CIIM convergence question as a classical fixed point problem in the local metric space of admissible analytic potentials. A further result exhibits a set of simple, natural conditions sufficient to guarantee that the CIIM operator is a self-map on a subspace of “normal” analytic potentials.